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MATHEMATICS LIBRARY 


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STANDARDIZED REASONING TESTS 
IN ARITHMETIC AND HOW 
TO UTILIZE THEM 


ByiClIREy Wa LONE. Ph.D: 


Director of Teaching, Iowa State Teachers College, Cedar Falls, Iowa 


PUBLISHED BY 
Gearhers College, Columbia University 
NEW YORK CITY 
1916 


Copyright, 1916, by CiirFr.W. STONE 


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A MATHEMATICS LIBRARY 


TABLE OF CONTENTS 


CHAPTER PAGE 
PEEL ETERODUCTION fers 2 fate oo iene OR Gd dk ete Oren 1 
The Test in Fundamentals 
The Test in Reasoning 


II. CONDITIONS AND DIRECTIONS FOR USING THE TEST............ 4 
The Main Points to be Observed 
How to Give the Test 
Directions to Pupils 


PASMIUIREOTIONS FOR SCORING nei iro cmenr yc Sy ely o e/uca, a a lane ae 6 
The Four Steps 
1. Marking the solutions 
2. Scoring the solutions 
3. Reducing scores to comparable bases 
4, Computing accuracy 
Directions for First Step 
Directions for Second Step 
The weightings for scoring 
Directions for Third Step 
Directions for Fourth Step 


IV. REPRESENTING, INTERPRETING, AND UTILIZING SCORES......... 9 
Scores of the Original Twenty-six Systems 
Accuracy of the Original Twenty-six Systems 
Representing Scores as Aids in Supervision 
The Butte, Montana, representation 
A graph form for representing percentages 
The Bloomington, Indiana, VI-A scores for 1914, with ex- 
planation of the procedure in making a graph 
A portrayal of the January, 1916, status of V-B, VI-B, and VII-B 
classes, Iowa State Teachers College Training School 
Record of Bloomington, Indiana, progress 
Record of Hackensack, New Jersey, improvement 
Representing Scores as Aids in Teaching . 
Scores of individual pupils 
Accuracy of individual pupils 
Both scores and accuracy of pupils 
Interpreting Scores 
Tentative standards 
Two Dangers 
Diagnosing causes of low scores 
Prescribing for improvement 


Equivalent tests VE 
lil 


OP fs 8 2 : 3 
ceS fi Re Senn! Ma 


CHARRE KE 
INTRODUCTION 


The occasion for publishing this booklet is the exhaustion of 
the edition of the book, ‘‘ Arithmetical Abilities,” in which the 
reasoning test herein contained first appeared. Then, too, it 
has been found that the brief record of the test as given in the 
original book does not give adequate help for those who may 
wish to utilize the test as a means of measuring the reasoning 
ability of children. It is hoped, then, that the publication of this 
booklet will furnish additional help to those who may wish to 
use this test both in supervision and in teaching. 

Another purpose of publishing this book is that of furthering, 
in some measure, progress in the scientific study of the teaching 
of arithmetic. This progress will doubtless move along two 
lines, viz., the development and use of better and more complete 
tests, and the settling of some of the many problematic conflicts 
in the teaching of arithmetic.’ 

The test in “ fundamentals ” or formal operations as embodied 
in the book, “ Arithmetical Abilities,’ has been so modified and 
improved by Courtis that it has been replaced by his Series B.' 
For the benefit of those who may wish to use it, the author’s 
original test in fundamentals is here reproduced and permission 
is given for it to be reprinted so that each pupil may have a copy. 


1 Stone, Cliff W., Arithmetical Abilities and Some Factors Determining 
Them. This book is out of print but may be borrowed from libraries of 
teachers colleges or schools of education. Extended extracts may be found 
in Strayer’s ‘“‘ Teaching Process’; Thorndike and Strayer’s ‘‘ Educational 
Administration’; Brown and Coffman’s ‘‘ How to Teach Arithmetic’; and 
Howell’s ‘‘A Fundamental Study in the Teaching of Arithmetic.”’ 

2 Some of these conflicts are stated in an article by the author entitled, 
Problems in Scientific Study of the Teaching of Arithmetic, Journal of Educ. 
Psychology, Vol. IV, No. 1, pp. 1-16. 

3 These tests may be secured from the author, S. A. Courtis, 81 Eliot St., 
Detroit, Michigan. 

Nt 


2 Standardized Reasoning Tests 


TEST IN FUNDAMENTALS 


Work as many of these problems as you have time for; work them in order 
as numbered. 


if Add 2375 
4052 

6354 

260 

5041 

1543 


2% Multiply 3265 by 20. 
Divide 3328 by 64. 
4. Add 596 
428 

94 

fie 

302 

645 

984 

897 


oe 


Multiply 768 by 604, 
Divide 1918962 by 543. 
7, Add 4095 
872 
7948 
6786 
567 
858 
9447 
7499 


Sa rek 


8. Multiply 976 by 87. 

9. Divide 2782542 by 679. 
IO. Multiply 5489 by 9876. 

1 Divide 5099941 by 749. 
Tae Multiply 876 by 70. 

13. Divide 62693256 by 8509. 
Td. Multiply 96879 by 896. 


Standardized Reasoning Tests 3 


The reasoning test is also reproduced here and those who 
wish to do so may have it reprinted so that each pupil may be 
provided with a copy. Those who prefer may secure copies, 
with a card of directions to pupils, in lots of fifty at the rate of 
50 cents per package. Address: Bureau of Publications, 
Teachers College, Columbia University. Prices of quantities 
quoted upon application. 

A second test, equivalent to the following, is being formulated 
and may be secured from the above address. 


REASONING TEST 


Solve as many of the following problems as you have time for; work them 
in order as numbered. 

1. If you buy 2 tablets at 7 cents each and a book for 65 cents, 
how much change should you receive from a two-dollar bill? 

2. John sold 4 Saturday Evening Posts at 5 cents each. He 
kept 14 the money and with the other % he bought Sunday 
papers at 2 cents each. How many did he buy? 

3. If James had 4 times as much money as George, he would 
have $16. How much money has George? 

4. How many pencils can you buy for 50 cents at the rate of 
25tor.5 cents? 

5. The uniforms for a baseball nine cost $2.50 each. The 
shoes cost $2 a pair. What was the total cost of uniforms and 
shoes for the nine? 

6. In the schools of a certain city there are 2,200 pupils; 
4 are in the primary grades, 44 in the grammar grades, % in 
the high school and the rest in the night school. How many 
pupils are there in the night school? 

7, If 3% tons of coal cost $21, what will 51% tons cost? 

8. A newsdealer bought some magazines for $1. He sold 
them for $1.20, gaining 5 cents on each magazine. How many 
magazines were there? 

9g. A girl spent 4% of her money for carfare, and three times 
as much for clothes. Half of what she had left was 80 cents. 
How much money did she have at first? 

10. Two girls receive $2.10 for making button-holes. One 
makes 42, the other 28. How shall they divide the money? 

11. Mr. Brown paid % of the cost of a building; Mr. John- 
son paid % the cost. Mr. Johnson received $500 more annual 
rent than Mr. Brown. How much did each receive? 

12. A freight train left Albany for New York at 6 o'clock. 
An express left on the same track at 8 o'clock. It went at the 
rate of 40 miles an hour. At what time of day will it overtake 
the freight train if the freight train stops after it has gone 56 
miles? 


CHAPTER II 


CONDITIONS AND DIRECTIONS FOR USING THE 
AW mae 


To get the greatest benefit from using a test the conditions 
under which it is given should duplicate those under which it 
was standardized. This test in reasoning is standardized in 
that it was given by the author to over three thousand pupils 
in one hundred and fifty-two classes of twenty-six representa- 
tive school systems of the United States. These measurements 
were made in 1907 and 1908, and the test is further standardized 
in that it has been subsequently given in a number of representa- 
tive school systems. 

In the schools tested by the author, and presumably in the 
subsequent tests by others, the conditions were under such con- 
trol that they were similar in each room of each school. A full 
statement of the precautions taken to maintain similar conditions 
is given in Part I of “ Arithmetical Abilities.” 

~The main points to be observed: 

1, No announcement that a test is to be given should be 
made to pupils. 

2. All directions to pupils should be given by the tester. 
(If the tester is a person from outside the system and 
if the teacher and principal are busy filling blanks with 
helpful data, the original conditions will be more fully 
duplicated.) 

3. Principals and superintendents should refrain not only 
from communicating with pupils about the tests, but 
also from being present in the room either immediately 
before or during the test. 

4. The time limit should be kept exactly—fifteen minutes 

to the second. 

No mention should be made to the pupils of time limit. 
Furnish each pupil with a copy of the test and have 
printed side turned down, until all begin. 


ae. 


Standardized Reasoning Tests 5 


7, Nothing should be said to pupils about the use of scratch 
paper, working in steps, amount of work to put down, etc. 


How to give the test: 

1. See to it that the conditions are a duplication of those 
stated above in so far as practicable. 

2. Arrange for the test with the teacher of the pupils. 
(If you do not know her, it is best to have a note of 
introduction from the principal or superintendent.) Ask 
her to kindly help you by answering a set of questions 
headed, Helpful Data furnished by the teacher at the 
time of the test.* 

3. Give either two or more of the Courtis tests in funda- 
mental operations or the Stone test in fundamentals. 

4. Give the reasoning test. If Courtis tests precede, use 
exactly the following directions to pupils.* If the Stone 
test in fundamentals precedes, proceed as directed in 
footnote (2). 


1. Take the materials that you usually take for your arithmetic 
work. Prepare two sheets of paper—headings and all. Have two 
sheets ready in case you may need them. Use pencils. Keep slip 
with printing turned down until we are ready to begin. 

2. Now, do you have everything ready? In order for you to do 
your best in this test, you will need to do just as all the other boys 
and girls who have taken this test have done. So pay close atten- 
tion and do just as I ask you to do. 

3. You will not need to mark these (slips) papers at all. You 
will find directions at the top of these (slips) papers, telling you 
just what to do, so you will not need to ask any questions and—I 
do not think I need to say this to you, but I will, just because I 
have to all the other boys and girls,—be especially careful not to 
see anybody else’s work. It is not easy not to see, but if you pay 
close attention to your own work only, the test will be the best. 

4. Begin. (Allow exactly fifteen minutes.) 


1These questions may cover such items as length of time devoted to 
arithmetic, methods employed, etc. For further suggestions write C. W. 
Stone, Iowa State Teachers College, Cedar Falls, Iowa. 
2If the Courtis tests are used, the directions contained in the folders 
accompanying them should be carefully followed. If the Stone test in 
fundamentals is given, follow directions given above under (4) except allow only 
12 minutes. And after collecting the papers on fundamentals say: 
1. Have two sheets prepared again. You may not need both but 
have them ready. 
2. Keep the printing turned down until we are ready to begin—the 
same as before. Now are all provided? 
3. Begin. (Allow exactly fifteen minutes.) 
3In order that these directions may be followed verbatim it is best to 
have them copied on a card, that the tester may hold them before him while 
giving the test. These cards may be had from the Bureau of Publications, 
Teachers College, New York City, at 5 cents each. 


CHAPTER 1s 
DIRECTIONS FOR SCORING 


In order to have the scores of a system, school, or pupil. 
comparable with those in the scale as standardized, the papers 
should be marked and scored as were those in the original study. 
To this end it is essential that each of the following steps be 
taken according to directions: 

(1) Mark the solutions on the basis of right or wrong 
reasoning. (2) Score the solutions according to assigned 
weighting and find scores for individual pupils. (3) Com- 
pute the total of the individual scores on the basis of a 
hundred pupils or compute the per cents of pupils making 
scores of 0, I, 2, etc. (4) Compute the per cents of accuracy. 


DIRECTIONS FOR First STEP 


In marking the solutions the following rulings should be ob- 
served: 

1. Mistakes in copying are not counted against the child; 
e.g., 


2X 7 = 14 
14and 65 = 
100 — 89 = il 
The reasoning of the problem was correct, so the mistake in copy- 
ing is not counted. 


2. Errors in fundamentals do not count against the reasoning 
score; €.g., 


$2.50 $5.50 
$2.00 9 
$5.50 cost of each. $50.50 


The reasoning in this problem is correct, so the mistakes in funda- 

mentals are not counted. 
3. The child is given credit for the part of the problem that 
is reasoned correctly when part is wrong—the amount of credit 
depending upon the number of steps reasoned correctly; e.g., 


6 | | | 


Standardized Reasoning Tests 7 


7 65 
2 —14 
14 41 


Since one of the steps is correct, this was counted 1/3 correct. 


4. If a problem is unfinished, credit is given for the steps 
taken if they are correct, or in so far as they are correct; e.g., 


(a) 2200 pupils. 
1/2 of 2200 is 1100 pupils in primary grades. 
This was counted 1/5 correct. 


(b) $2.50 cost of one uniform. 
9 


$22.50 cost of nine uniforms. 
$2.00 cost of one pair shoes. 


$18.00 cost of nine pair shoes. 
This was counted 2/3 correct. 


DIRECTIONS FOR SECOND STEP 
The weightings to be used in scoring the solutions are: 


For first problem, 1 
Seesecond i.) “ 1 
SOC iT & 1 
“fourth « 1 
ube Aa £ 1 
<Sesixth . 1 
“seventh “ : 
1 
2 
je 
2 
yy 


AN 


“eighth # 
oS rankthiel e 
“tenth s 
eleventh 
Peet wellthiens 


For example, any one of the first five problems solved cor- 
rectly should score 1 for the pupil, school, and system; the 
sixth would count 1.4; the seventh, 1.2, etc. In the third illus- 
tration under Directions for First Step, the solution of the first 
problems as given would count % for the pupil, school, and 
system; and in the first illustration under the 4th ruling the 
partial solution of the sixth problem would count 1/5 of 1.4 
or .28. 

The score of each individual pupil is found by adding all the 
scores of all the solutions found on his paper. 


8 Standardized Reasoning Tests 


DIRECTIONS FOR THIRD STEP | 

Sort the papers, placing the smallest score on top, and gradu- 
ate so that the largest is on the bottom. 

To get a single measure of a class, school, or system, add 
scores of all pupils and compute on basis of a hundred pupils; 
e. g., Mr. Hebden found the score for 716 Baltimore city pupils 
to be 5749.48. Reduced to the basis of a hundred pupils this 
is 803.2 as the score for the Baltimore system. (This is the 
measure used in the Springfield, Illinois, Survey and by Supt. 
Stark in the Hackensack Report. See graphs in Chapter IV.) 

To get the most helpful measure for supervision, compute the 
per cents of pupils that made scores of 0, 1, 2, 3, 4, etc., and 
construct a surface of distribution showing deviation from the 
standard. (This is the plan followed in the Butte, Montana, 
Survey and in handling the Iowa State Teachers College Train- 
ing School scores. See graphs in Chapter IV.) 

The most helpful measure for teaching is the score of each 
individual pupil. (Illustrations are shown at the end of Chap- 
ter IV in the representation of the scores of the VI-A-I class 
of the Bloomington, Indiana, schools and in the VI-B class of 
the Iowa State Teachers College Training School.) 


DIRECTIONS FOR FourTH STEP , 


The per cent of accuracy is found by computing the per cent 
reasoned correctly. In going over the papers for the data for 
this figuring, the following rulings should be observed: 

1. In determining the number of problems attempted, count 
number on which any work was done. This will be suffi- 
ciently exact for fifty to a hundred pupils. But for deter- 
mining the number of problems attempted by individual 
pupils, if the last problem worked is incomplete, use frac- 
tions to represent the steps attempted. 

2. In determining the number of problems correct for fifty 
to a hundred pupils, count the number that are worked 
entirely correct. But for computing the scores of indi- 
vidual pupils, use fractions to represent the steps correct. 


CHAPTER IV 


REPRESENTING, INTERPRETING, AND UTILIZING 
SCORES 


The degree of benefit derived from giving the test will turn 
on the effectiveness with which the results are utilized. One 
of the essentials for full utilization is effective representation. 
There are many effective means of representing scores and the 
one which a given person uses should be the one that is most 
effective for that person. For some, this will be tables of scores ; 
for others, surfaces of distribution; for others, graphs of pro- 
gress. Simple graphs are not difficult to construct and a little 
practice will help most persons to use them to great advantage. 
Whatever scheme is used the result should be that the scores 
are so placed that they may be readily and fully interpreted. 

Whether tabular or graph representation is used the standings 
of the twenty-six systems tested for the study, “ Arithmetical 
Abilities,” will be needed. Given by Roman numerals instead 
of by names these systems, with their scores, are: 


System XXIII, score 356 
“ ax “ 


XXIV 429 
A TP AEE “ 444 
£ IV “464 
SOX: Vi “« 464 
Non XO LL ebrigt Fete) 
ca Ae, GG | “ 469 
“ oe (13 49 1 
“aos VLE ei S09 
“ “ 532 
aka * §33 
Cua ELD ie Wet 
“ I “ 550 

Median “ 551 
“ I “ 552 
“ mE “ 601 
% II CraLO15 
oe he SL com O2¢ 
Sane oe EL w 4.050 


IO Standardized Reasoning Tests 


System XIV score 661 
“ exe “ 


691 
Dae ee Oe 
4 ale > SB TEL OO 
ee ee AY 
mp aww VAL ea aod 
hele Bibs s “848 
“ V “ 9 14 


The above tabulation is on the basis of a hundred pupils to 
each system and the first step in representing scores is to reduce 
them to that basis, e.g., seventy-five pupils in the Bloomington 
schools made a combined score of 539.65 in 1914. Reduced to 
the basis of a hundred pupils this is a score of 720. See graph 
of Bloomington scores, page 15, as an illustration of the use of 
this table. 

Accuracy in work is quite as important as amount of work, 
hence, besides the scale of scores showing the gross achievements, 
there is need of the scale of accuracy. The following is the 
table of per cents of problems correct in the twenty-six systems 
originally tested. See graph of Bloomington records, page 20. 

System XVI 54.9% correct 
. DG A ORES Bas £ 


PS eC LLL IbOO y Verena 
Shame LL 03 3.95 ine 


ee 65.3% 4 
. XV 66.3% - 
e VI 68.2% . 
a PO GND Tire bys 4 
“ 70.3% «“ 


Standardized Reasoning Tests II 


A simple though not especially effective representation is that 
of placing the score of a given system with that of other repre- 
sentative systems. The following is an illustration of this repre- 
sentation as used in the Springfield, Illinois, Survey: 


TABLE 23.—ScCORE PER EAcH 100 PuPpiILs IN REASONING IN ARITHMETIC IN 
SPRINGFIELD AND 26 OTHER SCHOOL SYSTEMS 


Spring- 
Spring- | field’s 
Lowest | Middle | Highest | field rank 


from top 
PR CO UNG Soy ees ee ai rel 356 550 914 508 19 
FM ae Vey eae AR a A 55 72 86 70 19 


The figures in the last column of the table show that in both 
the amount of work accomplished and the accuracy with which 
it was done the Springfield children rank in the 19th place 
among the 27 systems compared, ‘That is to say, they are more 
than two-thirds of the way down the list. 

The purpose of the representation is the best criterion for 
deciding which of the various schemes for representing to em- 
ploy. The main purposes for which scores are represented are 
to aid supervision and to aid teaching. Superintendents and 
other supervisors responsible for the work of many teachers 
need to be able to see readily the status of a class as a whole; 
teachers, on the other hand, need to be able to see readily the 
status of pupils as individuals. 


REPRESENTING SCORES AS AIDS IN SUPERVISION 


For purposes of supervision the ideal representation is that 
which conveys the class status with the minimum of time and 
effort. One of the best plans is that of the per cent distribution 
according to scores. This is illustrated in the portrayal of the 
Butte, Montana, reasoning scores (Fig. I). Fig. I represents 
the percentage of children making the given scores in reasoning 
problems. For example, 19 per cent of the fifth grade children 
made score of 0; 19 per cent made score of 1; etc. The lines 
representing the median scores for each grade tell about how 
many in each grade surpass the median scores for the grades 
above, and how many fall below the median scores for the 
grades below. 


I2 Standardized Reasoning Tests 


Resutts of Aritnmetic Tests ae 
PERCENTAGE oF Pupits ATTAINING GivEN SCORES 
Prosiems INVOLVING REASONING 
MEDIAN SCORES 


STH 6TH TrH 8TH 
ily ae 6 


VW“ StH Grave | 


g 7TH GRADE 
Y 


MLZ 


Ls 


PERCENTAGES OF PUPILS 


Ys — STH GRADE 


MMM LP OT a. 


6 78 9 10K! 12 13 1415 


wn 
(@) 
So 
Po) 
m 
ry) 
a or) 
Lar 9 
Pr 
co) ¢¥) 
> 
fa 


1c. I. Representing percentage of children making the given scores in 
asoning problems. Butte, Montana. 


Standardized Reasoning Tests 13 
Representation by graphs will be facilitated by using co- 
Ordinate paper of suitable rulings. 


The following form shows 
a good size. 


Copies of this form on coordinated paper may be 


secured from the Bureau of Publication, Teachers C yess 
Columbia University. 


~ 
~< 
it 
fy 
= ae 
Lo | 
hs 
iv 
ms 
fut 
ow 
se 
wo 
wy, 
“ 


DEIRENG LE PSR ENTS LS 10 ff i213 EIS 
Fie, II, 


Graph form for representing percentages of pupils attaining given scores. 


14 Standardised Reasoning Tests 


CY es & : 
pupils MRI NG 
Vavyious Se ores 


fi 
i; 


gb SEA CTS pubiry) 
“Blooming bw, 
$e Ura. 1914 


EE MES Alera | | Te 

Fic. III. Percentages of VI-A (75 Bloomington, Indiana) pupils making 
various scores. 

The procedure in filling out a form to represent the percentages 
of a given system or class may be explained in connection with 
Figs. II and III. Following the directions as previously given 
(page 8) the papers were sorted and the following table de- 
rived for the 1914 VI A pupils of Bloomington, Indiana: 


Number grading 0O— .5, 1= 1.3% of the 75 pupils. 
“ “ “ “ 


Soihs5, == 3 eee 

“ “ T5— Daye ees 1.3% «& “ “ “ 
“ “ 2,.5— Hey aes 0.0% “ “ “ “ 
“ “ 3.5— 475. 5s= 6.6% “ “ “ “ 
« Ha {ai 5.5, pe 9.3% TE 1 OG “ 
: Hi 5S" G5,) 14-18 16% ee en 
: HGS 7151418 OCiKe ee 
“ “ 7.5— S25. j= 9.3% “ “ “ “ 
“ «“ 8,5— 955; 4— 513% “ “ “ “ 
: ¢/\9,5--10.5,1 10-13 3p eouce eo uae 
: 710,511.25) 0) 7 0480p nome aera 
; “dd ebo12 45) tiled ton) Cems 
“ “ 12.5—13.5, 3= 4.0% “ “ “ “ 


By comparing this table with the graph it will be seen that 
one pupil or 1.3% of the 75 pupils whose score was O-.5 is 


Standardized Reasoning Tests 15 


represented in the lower left-hand corner of the form; the 
1.3% with a score of .5—1I.5, is represented next; the 1.3% 
with a score of 1.5—2.5 next; and as there were no pupils 
receiving a score of 2.5—3.5 there is no representation on the 
form; but 6.6% hada score of 4 (3.5-4.5) and the graph extends 
above 4 to 6.6; the 9.3%' with a score of '5 are represented above 
5, etc. The vertical broken line at 6.5 indicates the Standard. 
Thus the supervisor can see at a glance (1) the percentage of 
pupils attaining each score, and (2) the percentages that are up 
to or above Standard. 

Another way of representing the data for supervisory pur- 
poses is illustrated in the following graph of scores of the 
Bloomington, Indiana, schools where the author has tested the 
reasoning abilities of pupils for a series of years. 


[000 
70 --flighest_oq-2t_s yatems oe UE -<--- 
foo 
79° 
Geo 
. G0 
400 
(309 ImpR¥ement in yeawn nq 
‘$9to-1914 2 Bloweming con Ind! 
Loo Cas Cenmrvar. sahoot 
Me - MeAie “ 
(00 siima titan) Yl ia Goss 


PAs Avevage e%  ehree. 


1910 1dI2 1413 i 


Fic. IV. Bloomington, Indiana, Schools 


16 Standardized Reasoning Tests . 


va 
ren n sis 
] 
] 


I 
1 
! 


| 


40 41 f2 33 74 


OD ELE ETB EN TONE VY ES Le POY ate fae le AN Does 


Fic. V. Reasoning abilities Indiana State Teachers College Training 
School, January, 1916. This representation enabled the supervisor to see at 
a glance where the pupils of each of these classes stood in reasoning at the 


midyear and to know what percentage of each class needed special attention 
during the second semester. 


17 


Standardized Reasoning Tests 


OSNINOC WHY STIVINSWVYONDAS 


DILSWHLIUY NI SLSAL SNOLS 


Another markedly effective illustration is that employed by 


Supt. Stark of the Hackensack, New Jersey, schools 


. 3400 2S 
: D> 3dVYD ea +74 
A 16 Pat 
SIODHIS|) WOYSNIMovH 
Fees ES 
duos : oO <1 PHOS 
ee “389g HOS 
D a 
; AD BOvYY»D SJOoHeS PIv 
b G16) mgt 
SNOOHIS WOVSNIMOWH 
2 i=] = 
eoHI9 oi 
vlog ‘ ie 
YS Z0OVYD ih 
$s B2SA5 NOOHDS Y, | | 
HOWE Yos SBYOTS | HM 
"I 30 YI Vw {ipl o3¢ 34 7: 
“e - f Sra bens (] site fe ti 


4 


oo 


Fic. VI 


REPRESENTING SCORES AS AIDS IN TEACHING 
For purposes of teaching, the ideal representation is that which 


conveys the status of individual pupils with the minimum of 


18 Standardized Reasoning Tests 


time and effort. As far as the author has found, the best form 
of graphs for this purpose is that which shows the exact status 
of each individual pupil. As the supervisor needs to see by 
classes, so the teacher needs to see by individuals. The follow- 


ing are illustrations: 


ie Svoves: Sk yn 

jap TVA 4A Pupils. 

WE-R-1y Bloomington 

A Fuad. 3 : 


~ 


Stendayte pL LEE EL Lt mies 


Se ee NS OS ONS 8 


O=TSin elo folate a arshaters potatststoror 
PISS ISIE Slate isis lalshiaiak 


_._ Fic. VII, Each line stands for the score of a pupil. For purposes of 
identification the initials of the pupils are placed at the bottom, e.g., J.H. 
not only reached Standard, but did so well that his line extends almost twice 
as high as the Standard of the VI A Grade requires. This form of graph 
brings out vividly just where each pupil stands, e.g., E.L. with a score of 
less than 1. 

Coérdinated paper enables one to make such graphs readily and accurately. 


Standardized Reasoning Tests 19 


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Fic. VIII. Here the lines stand for per cents of accuracy for individual 
pupils, e.g., D.C. is well above the Standard in accuracy, while E.L. is far 
below. 


INTERPRETING SCORES 


The scores having been represented effectively, the next im- 
portant step in the use of the test is now possible, viz., that of 
interpreting the results, This is preéminently the responsibility 
and privilege of the teacher. To do it successfully requires the 
most intimate knowledge of the individual pupils. 

The interpretation of results implies an agreement as to 
standards, and unfortunately this is one of the unsolved prob- 
lems of education. Probably one of the most satisfactory bases 


20 Standardized Reasoning Tests 


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Fic. IX. This is a representation of both scores and accuracy for indi- 
vidual pupils. It enables the teacher to see at a glance the status of each 
pupil in both amount and quality of work, e.g., J.H. is seen to be above 
Standard in both, while L.D. is markedly accurate but does not reach the 
Standard in score. This shows the teacher that L.D.’s difficulty is lack of 
speed and she is in a position to determine the cause of the slowness and to 
set about removing it. 


now possible for proposing standards is the achievements of 
the better schools of the United States. Whether teachers 
should consider their pupils up to standard if 50 per cent of them 
reach or surpass the median ability shown by the twenty-six 
representative systems or whether they should strive to have 
80 per cent of them attain that ability, is a question which the 
school workers of each system must settle for themselves. The 
important point is that such an agreement as to standards is 
now possible and that it is very beneficial. The author offers 
the following as tentative standards for grades V to VIII. 


Standardized Reasoning Tests 21 


TENTATIVE STANDARDS 


That 80 per cent or more of 5th grade pupils reach or exceed 
a score of 5.5 with at least 75 per cent accuracy; that 80 per 
cent or more of 6th grade pupils reach or exceed a score of 6.5 
with at least 80 per cent accuracy; that 80 per cent or more of 
7th grade pupils reach or exceed a score of 7.5 with at least 
85 per cent accuracy; that 80 per cent or more of 8th grade 
pupils reach or exceed a score of 8.75 with at least 90 per cent 
accuracy, 

Having decided where pupils ought to be, the results of the 
tests will show how many of them are there and then it remains 
for the teacher and her supervisor to decide (1) how many 
others can and ought to progress towards that goal, (2) how 
far each can probably go, (3) how best to help them in that 
progress, and (4) how best to help those who have arrived to 
make the best use of their time. 


Two DANGERS 


In interpreting the results of a test, care should be taken to 
avoid two dangers: (1) that of placing too much reliance on a 
single measure of an individual. (No pupil whose score is sur- 
prisingly low should be so graded without a second trial. The 
first test may have come on an “off day” for the individual. 
See Fig. X); (2) that of relying on the average of class scores 
to show whether the pupils are “up to standard.” (It may be 
that there are a few very high scores and a few very low scores, 
and both these undesirable extremes will be hidden in the aver- 
age. It is always best to study the scores of individual pupils 
and if a single measure of ability is needed, to use the median.) 


DIAGNOSING CAUSES OF Low SCORES 


The causes of failure or success are many and often complex. 
In the results of the mental processes of reasoning it is especially 
difficult to be certain of the exact causes. In diagnosing cases 


22 


Standardized Reasoning Tests 


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Fic. X. The score of B. H. was unexpectedly low and to have diagnosed 
her case on the basis of the first test would have been inaccurate and unjust. 


of failure the following are some possible causes that may well 
be considered: 


I. 


i) 


Inability to read——Any pupil who is poor in reasoning 
should be thoroughly tested by the use of some of the 
measures of silent reading. ! 

Poor judgment as to the amount of writing to do in the 
solution.—Much time and energy are lost by labored labels 
and elaborate indications of steps. 

Physical disability—Poor eyesight is a common hindrance 
in this as well as other phases of school work. 

Lack of physical codrdination.—Tests in copying figures 
and other coordination tests should be used. 

Inability to see relationships between steps.—This is fre- 
quently an innate lack and very difficult to overcome. 
Mental laziness.—There is no spur equal to knowing just 
where one stands and where one ought to expect to stand. 
Lack of a realization of the passing of..time. 

Temporary disability—Individuals have “off days” and 
surprisingly low scores should not be regarded as the meas- 
ure of an individual without a second trial.* 


1 But as the first scores are counted in making the scale, it is best that 
all pupils be included in the scores as computed for the class or system. 


Standardized Reasoning Tests 23 


PRESCRIBING FOR IMPROVEMENT 


A record of the treatment of certain individuals! with low 
scores will serve to illustrate the possibility of prescribing for 
improvement. The records will be given under diagnosis, treat- 
ment and results. 


Pupil, H. C. 


Diagnosis: Up to standard in reading ability, did not indulge in undue 
labeling, physical examination showed no defects, constantly made low 
scores. Conclusion as to cause of low score: Mental laziness with lack of 
realization of the passing of time. 


Treatment: The pupil was first of all made conscious of his status by 
comparing his score with those of his fellow classmates and with the stand- 
ard; then he was helped to study his way of working which convinced him 
of the seat of his difficulty. From day to day lists of approximately equiva- 
lent problems were given him with time limit. Much was made of record of 
scores, gain being expected by both teacher and pupil. 


Results: Within a few days notable gain appeared, due to increased 
ability to direct and hold attention to the work in hand. Contrasted with 
his previous tendency to wander, the pupil became capable of working con- 
tinuously in spite of such distractions as people entering the room. After 
about twenty minutes daily for three weeks he raised his score from 4 to 
5.4. Though this is not a large gain in score, the boy had made it largely 
of his own initiative; he had formed an ideal of concentration, and the con- 
cept of giving attention to reasoning processes was well under way. It is 
believed by those who have studied the boy that much of his improvement 
was due to the convincingness of the objective evidence of his need to improve. 


Some Pupils of a Certain Fifth Grade 


Diagnosis: Many pupils made very low scores, many papers much cov- 
ered with such statements as, ‘‘ If one tablet cost 7 cents, 2 tablets 
etc.’”’ Here was evidently one large source of failure. 


Treatment: Emphasis was placed on the possibility of saving time by not 
writing so much, brief labels were devised, originality was encouraged, and 
approval of pupils and teacher placed on briefest adequate statement. 


Results: As shown by second test and by daily work, much time was 
saved for reasoning processes. The following parallel columns show typical 
results. 


Pupil, A. K. 
In first test In second test 
They would cost $18. $2.50 XK 9 == $22.50 
If one suit cost $2.50, 9 would cost $2 < 9 = $18. 
$2.50 * 9 = $22.50. 
They would cost $40.50. $40.50. 
Score in first test, 1 1-3. Score in second test, 3, 


1This record of pupils was furnished by Miss Floe E. Correll, supervising 
critic of mathematics, Iowa State Teachers College Training School. 


24 Standardized Reasoning Tests 


Pupil, LTC. 
In first test In second test 
If he sold 4 papers and got twenty 5 One-half would be 
cents for them, one-half would be 10 4 10 cents and he 
cents and with the other 10 cents he — could buy five. 
bought Sunday papers, he would buy 20 


as many as 2 will go into 10 or 
2|10 

5 papers. 
Score in first test, 3, Score in second test, 4 1-2, 


EQUIVALENT TESTS 


Equivalent tests are desirable for measuring the results of 
treatment or lapses of time. Reasoning tests equivalent to the 
one included in this book, are being constructed by the author. 
They will be published and made available by Teachers College 
Bureau of Publications. Special care is being taken to have 
these tests strictly equivalent to the original; and the procedure 
in utilizing them will be the same as set forth in this book for 


the original test. 


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